College Algebra, 2013a

388 Further Topics in Functions
6
5
y
4
3
y = x
2
1
−6 −5 −4 −3 −2 −1 1 2 3 4 5 6
−1
x
−2
−3
−4
−5
−6
y = g(x) andy = g −1 (x)
We now return to f(x) =x 2 . We know that f is not one-to-one, and thus, is not invertible.
However, if we restrict the domain of f, we can produce a new function g which is one-to-one. If
we define g(x) =x 2 , x ≥ 0, then we have
y
y
7
6
5
4
3
2
1
7
6
5
4
3
2
1
−2 −1 1 2
x
y = f(x) =x 2 restrict domain to x ≥ 0
−−−−−−−−−−−−−−−→
−2 −1 1 2
y = g(x) =x 2 , x ≥ 0
The graph of g passes the Horizontal Line Test. To find an inverse of g, we proceed as usual
y = g(x)
y = x 2 , x ≥ 0
x = y 2 , y ≥ 0 switch x and y
y = ± √ x
y = √ x since y ≥ 0
x